To solve for the slant height (the distance from the apex of the pyramid to each vertex of the base), we'll follow these steps:
1. Identify the given values:
- Each side of the square base: 6 units
- Altitude of the pyramid: 10 units
2. Calculate the length from the center of the base to a vertex:
- The base is a square, so we first find the diagonal of the square base.
- The diagonal of the square [tex]\( \text{d} = a \sqrt{2} \)[/tex], where [tex]\( a \)[/tex] is the side length of the square.
- Therefore, diagonal [tex]\( \text{d} = 6 \sqrt{2} \)[/tex].
3. Calculate half of the diagonal to find the distance from the center of the base to a vertex:
- Half of the diagonal [tex]\( = \frac{6 \sqrt{2}}{2} = 3 \sqrt{2} \)[/tex].
4. Use the Pythagorean theorem to find the slant height:
- We have a right triangle where:
- One leg is half the diagonal: [tex]\( 3 \sqrt{2} \)[/tex]
- The other leg is the altitude of the pyramid: 10 units
- The hypotenuse is the slant height, which we are trying to find.
- According to the Pythagorean theorem:
[tex]\[
(\text{slant height})^2 = (3 \sqrt{2})^2 + 10^2
\][/tex]
[tex]\[
(\text{slant height})^2 = 18 + 100 = 118
\][/tex]
[tex]\[
\text{slant height} = \sqrt{118} \approx 10.9
\][/tex]
Therefore, the slant height of the pyramid, to each vertex of the base, rounded to the nearest tenth, is [tex]\( \boxed{10.9} \)[/tex] units.
